Proving an Equality: False Premise to True Result

[homology]

I was surprised when a few months ago, while talking to a fellow student, he suggested that the way you prove an equality (like P=Q) is you start with P=Q and play with it until you get something that's true, then you "know" that P=Q is true.

Now this is rubbish of course, since a false premise can imply a true one. And I showed him the example:

2=1 subtract 1 from both sides
1=0 add 1 to the left and 2 to the right to get
2=2

But he scoffed and said, "sure, sure" but you're using what you're trying to prove (the fact that 2=1). Well its clear that I haven't made him a believer, I was wondering if any folks here had really juicy examples of trying to prove P=Q, a false statement and ending up with R=S, a true one.

Thanks a lot,

Kevin

 

[amcavoy]

I was surprised when a few months ago, while talking to a fellow student, he suggested that the way you prove an equality (like P=Q) is you start with P=Q and play with it until you get something that's true, then you "know" that P=Q is true.
Now this is rubbish of course, since a false premise can imply a true one. And I showed him the example:
2=1 subtract 1 from both sides
1=0 add 1 to the left and 2 to the right to get
2=2
But he scoffed and said, "sure, sure" but you're using what you're trying to prove (the fact that 2=1). Well its clear that I haven't made him a believer, I was wondering if any folks here had really juicy examples of trying to prove P=Q, a false statement and ending up with R=S, a true one.
Thanks a lot,
Kevin

Let a=b

1 = 2

b = 2b

b = a+b

b(a-b) = a²-b²

ab-b² = a²-b²

ab = a²

This is working backwards (since you wanted to start with a false statement) an example from MathWorld.

http://mathworld.wolfram.com/Fallacy.html

 

[HallsofIvy]

I was surprised when a few months ago, while talking to a fellow student, he suggested that the way you prove an equality (like P=Q) is you start with P=Q and play with it until you get something that's true, then you "know" that P=Q is true.

Actually, that's a quite common method of proof, sometimes called "synthetic proof" and is often used in proving trigonometric identities: you start with identity (what you want to prove) and reduce it to something you know to be true.

homology is, of course, completely correct that a false statement can lead to a true one. You have to be careful that every step in a "synthetic proof" argument is reversible. What you are really doing is using a common method of deciding how to prove something- working backwards.
"Here is what I want to prove- what do I need to have so that that is clear? Okay, now what do I have to have in order to prove that?", continuing until you arrive at something you know to be true- a definition or axiom or a "given" part of the hypothesis. Having determined how to prove, you turn around and do everything in reverse- start with the "given" and work back to what you wanted to prove. As long as every step is reversible you can do that. If, in a simple proof, it is clear that every step is reversible, it may not be necessary to actually write out the "reverse" process- that's a "synthetic proof".

 

[homology]

Hmm,with all due respect, it seems fishy to me. I have to say that I would never prove anything by first assuming it and then working from there. While I might play with such things on scrap paper, a final proof should start with what is known to be true and then by deduction arrive at the goal.

Could you direct me to a rigorous definition of synthetic proof?

Thanks,

Kevin

 

[Robokapp]

Proving something is equal is much harder than proving it wrong. So...proving that 1=0

1^0=0^1 Wrong

1/0=0/1 Wrong

1-0=0-1 Wrong

(x-1)(x-0)=0
x^2-x-0=0 I'm completing the square
x^2-x-1/4-0+1/4=0
(x^2-x-1/4)+1/4=0
(x-1/2)^2+1/4=0
(x-1/2)^2=-1/4
(X-1/2)=+ and - 0.5i

so the equation formed by turning the given x values (0 and 1) into factors (x-1) and (x-0) does not have identical roots, therefore the factors must differ.

and you can keep on going like this as far as you want.

 

[eNathan]

I was surprised when a few months ago, while talking to a fellow student, he suggested that the way you prove an equality (like P=Q) is you start with P=Q and play with it until you get something that's true, then you "know" that P=Q is true.
Now this is rubbish of course, since a false premise can imply a true one. And I showed him the example:
2=1 subtract 1 from both sides
1=0 add 1 to the left and 2 to the right to get
2=2
But he scoffed and said, "sure, sure" but you're using what you're trying to prove (the fact that 2=1). Well its clear that I haven't made him a believer, I was wondering if any folks here had really juicy examples of trying to prove P=Q, a false statement and ending up with R=S, a true one.
Thanks a lot,
Kevin

If you have

$$a = b$$

you can't add different numbers to the left and right, you have to add the SAME number, or subtract, multiply, divide...ect

you can't add 1 to the left and 2 to the right!

nice try, however.

 

[DeadWolfe]

If you have
$$a = b$$
you can't add different numbers to the left and right, you have to add the SAME number, or subtract, multiply, divide...ect
you can't add 1 to the left and 2 to the right!
nice try, however.

By hypothesis, 1 and 2 are the same number.

 

[matt grime]

how about -1=1 and sqaure both sides?

the synthetic idea is quite easy to understand and it essentially saying that if two things are equivalent then it doesn't matter which side you start working from to get the answer, as long as all your "if then" deductions are in fact "if and only if". there is nothing wrong with that in any absolute sense though i find it distasteful if it is done in a bad way and unnecessarily. Too often i see people work it through backwards and then not check that all steps are reversible.